Vector-valued statistics of binomial processes: Berry-Esseen bounds in the convex distance

Understand

We study the discrepancy between the distribution of a vector-valued functional of i.i.d.

• random elements and that of a Gaussian vector.
• Our main contribution is an explicit bound on the convex distance between the two distributions, holding in every dimension.
• Such a finding constitutes a substantial extension of the one-dimensional bounds deduced in Chatterjee (2007) and Lachi\`eze-Rey and Peccati (2017), as well as of the multidimensional bounds for smooth test functions and indicators of rectangles derived, respectively, in Dung (2019), and Fang and Koike (2021).